climleltrp - Mathbox for Glauco Siliprandi

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Theorem climleltrp 45597

Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
climleltrp.k	⊢ Ⅎ𝑘𝜑
climleltrp.f	⊢ Ⅎ𝑘𝐹
climleltrp.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climleltrp.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
climleltrp.r	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
climleltrp.a	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climleltrp.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
climleltrp.l	⊢ (𝜑 → 𝐴 ≤ 𝐶)
climleltrp.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)

**Assertion**
Ref	Expression
climleltrp	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹 𝑗,𝑁,𝑘 𝑗,𝑋,𝑘 𝑗,𝑍 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑘) 𝐶(𝑗,𝑘) 𝐹(𝑘) 𝑀(𝑗,𝑘) 𝑍(𝑘)

**Proof of Theorem climleltrp**
Step	Hyp	Ref	Expression
1		climleltrp.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		climleltrp.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2854	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		uzss 12926	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
6	5, 2	sseqtrrdi 4060	. 2 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
7		climleltrp.k	. . . 4 ⊢ Ⅎ𝑘𝜑
8		climleltrp.f	. . . 4 ⊢ Ⅎ𝑘𝐹
9		uzssz 12924	. . . . 5 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
10	9, 3	sselid 4006	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11		eqid 2740	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
12		climleltrp.a	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
13		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘))
14		climleltrp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
15	7, 8, 10, 11, 12, 13, 14	clim2d 45594	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
16		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ_≥‘𝑁)
17	7, 16	nfan 1898	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁))
18		simplll 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝜑)
19		uzss 12926	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
20	19	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
21		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
22	20, 21	sseldd 4009	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑁))
23	22	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑘 ∈ (ℤ_≥‘𝑁))
24		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
25		climleltrp.r	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
26	13, 25	eqeltrd 2844	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℝ)
28		climcl 15545	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
29	12, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
30	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝐴 ∈ ℂ)
31	26	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	pncan3d 11650	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) = (𝐹‘𝑘))
33	32	eqcomd 2746	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
34	33	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
35	34, 27	eqeltrrd 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
36		climleltrp.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
37	36	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐶 ∈ ℝ)
38	7, 8, 11, 10, 12, 25	climreclf 45585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℝ)
40	27, 39	resubcld 11718	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
41	37, 40	readdcld 11319	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
42	14	rpred 13099	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	42	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑋 ∈ ℝ)
44	37, 43	readdcld 11319	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + 𝑋) ∈ ℝ)
45		climleltrp.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
46	45	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ≤ 𝐶)
47	39, 37, 40, 46	leadd1dd 11904	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ≤ (𝐶 + ((𝐹‘𝑘) − 𝐴)))
48	31	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℂ)
49	30	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℂ)
50	48, 49	subcld 11647	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
51	50	abscld 15485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
52	40	leabsd 15463	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
53		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
54	40, 51, 43, 52, 53	lelttrd 11448	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) < 𝑋)
55	40, 43, 37, 54	ltadd2dd 11449	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
56	35, 41, 44, 47, 55	lelttrd 11448	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
57	34, 56	eqbrtrd 5188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) < (𝐶 + 𝑋))
58	27, 57	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
59	18, 23, 24, 58	syl21anc 837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
60	59	adantrl 715	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
61	60	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
62	17, 61	ralimdaa 3266	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
63	62	reximdva 3174	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
64	15, 63	mpd 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
65		ssrexv 4078	. 2 ⊢ ((ℤ_≥‘𝑁) ⊆ 𝑍 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
66	6, 64, 65	sylc 65	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 Ⅎwnfc 2893 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 + caddc 11187 < clt 11324 ≤ cle 11325 − cmin 11520 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 abscabs 15283 ⇝ cli 15530

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535

This theorem is referenced by: smflimlem2 46693

W3C validator